Translate

Google+ Badge

Follow by Email

Search This Blog

Wednesday, January 11, 2012

I
used the word bond(s) quite frequently in the last chapter, and by now you
should have some grasp of what they are:when two atoms approach closely enough, the most stable state of their
outermost, or valence, electrons usually
centers between the atoms.Whenever two
or more atoms can bond this way, the resulting structure is called a molecule;
i.e., two hydrogen atoms and an oxygen atom will bond together to make the
stable water molecule, of H2O.We were also introduced to the idea that different atoms can form
different numbers (and types) of bonds:one for hydrogen, four for carbon, three for nitrogen, two for oxygen,
and so forth.(The four bonds carbon can
form will prove most convenient in coming chapters.)

There
are numerous ways of portraying bonds.In first year college chemistry courses the student is usually
introduced to the Lewis dot structure.Whereas before we represented bonds as bars
between atoms, e.g., H-H,
a better representation is to show the electrons themselves and where they are
located in space:H˸H.Here it is clear that the two hydrogen atoms
share their electrons in a region focused between the two nuclei.You can see the stability this creates, and
so why the bond is stable.

You’re
already suspicious though, I’ll bet.The
Lewis dot structure still commits a cardinal sin when portraying electrons; for
they are not hard little dots or spheres but spread out regions of uncertainty.Remember how in the quantum world, we cannot
say that objects cannot have exact locations, speeds, and directions; that if
we do devise an experiment to measure one quantity exactly, we lose all
information about the others.Probability distributions rule the day.Thus, an improvement on the dot structure representation would look
something like this:

Figure XIV.

Here,
the black spheres represent the hydrogen nuclei (usually just a proton), and
the smeared out red region the electrons in the bonded state.Note that we don’t even distinguish between
one electron and the other, only that their combined region of highest density
(probability distribution, that is) is directly between the nuclei.This is about the best representation of how
electrons behave in atoms and molecules.To answer your next question, the dotted line surrounding the molecules
encompasses ninety percent of the electrons’ distribution – in fact, this goes
off to infinity.

***

What’s
coming is, I think, rather grimmer than the material covered so far, but is so
essential to describing atoms and molecules and bonds that I see no way of
avoiding it.Some advanced equations
will be presented, but don’t worry, because there’s no need to understand them
or know how to solve them.I display
them for you so that you will see there is a mathematical underpinning to all
this talk of electron states and bonds.Truth be told of course, there is a mathematical underpinning to all
physics, so this should not come as a surprise.

But
first, a switch in terminology.Instead
of states we will use the word orbital
(both atomic and molecular) to describe an electron’s characteristics.Orbital is, in my opinion, an unfortunate
choice of word, as it implies electrons orbiting the nucleus, a fallacy I have
struggled to help you overcome.But it
has been used so long now that we are apparently stuck with it; just remember
that it is a synonym for state, not orbit, and you should be fine.With this in mind, let’s plunge into the
fray:

Figure XV.

This
is the Schrödinger equation for the hydrogen atom.You may recognize or know some of the symbols
it contains, but the one I want you to concentrate on is y,
which is the equation’s solution.Interestingly, it is a wave function, i.e., a function that describes
wave motion, such as that of a piano string vibrating back and forth.The general form of this function, the
solution to the equation, is:

Figure XVI.

Again,
don’t worry about most of the details; deriving this solution requires some
heavy duty math.There are certain
variables you should pay heed to, however.These are n,
ℓ, and m,
known respectively as the principle quantum number, the angular momentum
quantum number, and the magnetic moment quantum number.The combined values of these numbers
determines the exact form of y.There are relationships among them as well:while n
can have any integer value from one on up (1, 2, 3, 4, 5, … ¥, ℓ
can range only from 0 to n – 1, and m
from – ℓ to + ℓ.So for example, if n = 1 (the lowest
energy states of the atom), then l and m can only be 0; while for n = 2 ℓ
can have the values 0 and 1, while m
be one of three values, -1, 0, and 1.

The orbitals I mentioned earlier are
determined by their quantum numbers.The
number specify their shapes, the probability distributions of the electrons
occupying them (a maximum of two, known as the Pauli Principle), and the
energies of those electrons:

Figure VII. (repeated)

The
elements in the periodic table are built up by adding additional protons (and
neutrons) to the nucleus, and then filling the atomic orbitals from inwards to
outwards with the required electrons to maintain the atoms electrical
neutrality (atoms that are not neutral are called ions, but they are still
atoms).For the working chemist the
electrons are described by orbital designations; say, for selenium, or element
34, the designation would be 1s22s22p63s23p64s23d104p4,
where 1, 2, 3, and 4 represent the shell or primary quantum number, s, p, and d
the ℓ
numbers in the shell, and the exponent the number of electrons occupying each
orbital group in a shell (thus, there are 3
p orbitals, 5 d orbitals,1
s orbital, and so on).I won’t describe this scenario in any greater
detail because:a) it still befuddles me
some after all these years and I doubt I could describe it clearly and simply;
andb) our main thrust will be bonding/molecular orbitals, and only a basic
grasp of the atomic kind is needed to do so.

***

There
is one more revelation about atoms that must be made before turning to
molecules.In truth, all the atoms that
have been built up in the periodic table should be analyzed by constructing and
solving a Schrödinger-type equation for each one, yet we have only done so for
hydrogen and assumed the results applied to all atoms.Amazingly, and fortunately, they almost
do!In fact, we could set up an equation
for each elements; could set up, but never solve by any known mathematical
technique known to Homo sapiens.The reason?In hydrogen we are dealing with only two interacting bodies, but for
every other atom and molecule, we are working with more than two.The so-called three-body equation is intractable to all math and physics; not
just in the quantum world but everywhere, at all times, in the solar system and
the universe as a whole – though whether it explains why there are only two
sexes I have no idea:nature seems to
have no difficulty with it, but then nature is just doing what it does
according to its laws; it (She?) isn’t trying to calculate anything in advance.

Returning
to the equation of the hydrogen atom we are lucky indeed that its solution
gives us a sound working model for the other atoms as well; only minor
modification are needed to obtain excellent results for them.However, and this is important to emphasize,
any two or more solutions (functions) to the equation that have approximately
equal energy and obey the mathematical requirement for “orthogonally” can be
combined to make new functions/solutions.For example, on carbon the single 2s
and three 2p
orbitals can be combined to create four new “sp3“
orbitals, which stick out at tetrahedral angles from the carbon nucleus:

Figure XVII.

I
suspect I have presented you with a puzzle.Why not use these orbitals to hold carbon’s four valence (outer)
electrons, instead of the original one from solving the Schrödinger
equation?The answer is simply that the
original orbitals are at lowest energy when considering lone carbon atoms; only
when chemically bonded to other atoms – including other carbon atoms – is this
configuration preferable, for reasons we shall see soon enough.

This
is not the only possible combination.Another “hybrid” (as they are know) orbital is the sp2
orbital, in which one of the p
orbitals – technically the pz)
is left out:

Figure XVIII.

Notice
that the omission one a p
orbital has changed the geometry of the atom from tetrahedral to a flat
triangle; the new hybrid orbitals comprising the triangle is called, reasonably
enough, sp2.Other hybrids or combinations produce a wide
variety of configurations, especially when they involve d or higher
orbitals.

***

I
made the blithe statement that all of this has to do with bonding with other
atoms to produce molecules, and it is time to explain why.

Atoms
takes up considerable space in molecules, and therefor can not simply assume
any arrangement.A good example of this
is the compound methane, or CH4.We should like to spread the hydrogen atoms
and their electrons as far apart as necessary, in order to reduce interelectron
repulsions.As a first guess we could
try a flat structure:

H

│

H¾C¾H

│

H

Figure XIX.

That
this is inadequate should not be difficult to see.We have here a two dimensional representation
of methane; and although it is often convenient to draw it out this way, the
lack of a third dimension means that the hydrogens are not spread as far apart
as possible.

Go
back to the tetrahedral arrangement of electrons around carbon, and you can see
that the carbon’s electrons and the hydrogen’s electrons finally put as much
space between each other as possible, to produce the configuration with lowest
energy:

Figure XX.

Comparing
this with Figure XVII.
We see that the tetrahedral arrangements of hydrogens about the central carbon
exactly matches its 3p3
bond arrangement.This is why these are
the preferred orbital pattern in the molecule.The only thing left to do in our preliminary exploration is to describe
just what a bond is.I’ve used an
equivalent phrase molecular orbital, but with out explanation what that means.

The
fact that I’ve used bond and molecular orbital interchangeability should be
suggestive however, implying that bonds are orbitals too, and this is quite
right.Just as atoms can combine their
base orbitals into hybrid orbitals, so can orbitals on different atoms also merge into molecular orbitals, given the right
conditions of fairly equal energy and orthogonality.Remember the depiction of the hydrogen
molecule earlier in the chapter:

Figure XIV.
repeated

Of
course, we really need to solve the Schrodinger equation for the system, but we
can’t so we must find another way.Again, that is the linear combination of orbitals technique.The 1s orbitals on each hydrogen atom add
together to produce the molecular orbital or bond, within which the two
electrons have even lower energy than single atoms.Bear in mind, though, that the number of
hybrid or molecular orbitals must equal the number of base orbitals we began
with.Thus there are four sp3
orbitals about carbon (and nitrogen and oxygen, and many of atoms).So combining the 1s
orbitals on the two hydrogens must yield two new orbitals.And indeed they do; in addition to the
bonding orbital, there is also an “anti-bonding” orbital, an orbital which has
the effect of repelling the hydrogens if there are electrons in them as much as
the occupied bonding orbital which attracts the hydrogen:

Figure XXI.

Notice
that the main electron density is on the outside of the atoms instead of
between them in Figure XIV.Some things ought to be
starting to gel by now.Normally,
hydrogen exists as a diatomic molecule, H2.This is because the two electrons from the
atoms can fill the bonding orbital, but the anti-bonding are empty.On the other hand, the next element in the
periodic table, helium or He, exists a monoatomic atoms.He2
, if it existed, would have two more electrons than H2,
and those two electrons would fill the anti-bonding orbital, pushing the
heliums apart and breaking the bond.Hence,
only He atoms are found in nature.

***

There
is one more step until we can start letting the horses out of the barn.We’ve spoken so far of base atomic orbitals
(1s,
2p,
3d,
etc.), hybrid atomic orbitals (sp3andsp2),
and molecular orbitals resulting from combining base atomic orbitals.Look back on the picture of the methane
molecule (Figure XX).Can you see how the sp3
orbitals on carbon are combining with the 1s
on the four surrounding hydrogens?Remember, any proper linear combination of orbitals (strictly speaking,
their underlying wave functions) yields new orbitals.So hybrid orbitals can combine with atomic,
other molecular, or hybrid orbitals.The
possibilities are mind-boggling, perhaps even infinite, especially as you add
more and more atoms to the potential molecules.Many millions of molecules are already known to science.Obviously, I can only cover a very tiny
fraction of them, but fortunately that will do to get the idea in your mind
clearly.

Let’s
look at methane again, and try an experiment in our minds.What if we were to take a methane and pop off
one of the hydrogens?Why then, we would
have something the looks like this:

H

│

H¾C·

│

H

Figure XXII.

Forgive
the flat structure, which we know not to be true, but there is method in
madness here.The first thing the hits
your eye is that the right hydrogen has been replaced by a dot, which
represents an unbonded electron.In
removing the hydrogen with its lone electron, we leave carbon’s contribution to
that bond hanging out there in space, so to speak.More importantly, it is now available for
bonding with another atom or molecule with a lone electron. For example, the
molecule here is actually called a methyl radical, radical meaning containing
unbonded electrons.We can certainly
combine it with another methyl radical:

HH

││

H¾C¾C¾H

│ │

HH

Figure XXIII.

Using the flat structure for methane makes it much
easier to draw this new molecule, called ethane.The bonds about each carbon are still
tetrahedral, however.

Incidentally, bonds of this nature, with head-on
overlap of the constituent orbitals are known as sigma (σ)
bonds.The way this chapter has
progressed, you can be excused for thinking it is the only kind of bond.There are several more, and right now I’d
like to concentrate on a specific one:the pi (π) bind.In keeping with methane and ethane, I’ll show
you the simplest case, that of ethylene:

Figure XXIII.

Before discussing the “double” bond between the
carbon atoms in ethylene, I want to draw your attention to the flat, trigonal
geometry about them.We have already
seen this, in Figure XVIII,
the sp2
hybrid orbitals which creates this geometry.Four of these orbitals combine with the 1s
orbitals on the hydrogens to create those bonds, while the two carbon pointing sp2
overlap to create a sigma bond, just as with H2.But this leaves something unaccounted
for.Both carbon’s pzorbitals
are now free to make other combinations, and each contains a single electron it
can use for bonding.In fact, the most
stable bond it can form is by a sideways
overlap of the
pz orbitals.Such resulting bonds are called pi bonds, and
they look like this:

Figure XXIV.

Imagine
the two orange balloons on the left as the carbons’ pzorbitals;
then the yellow region on the right is the region of maximum overlap or
combination.This orbital (although
there are two lobes, it is one orbital, above and beneath the sigma, represented by the straight line) is the
bonding; the ps
also combine to make an anti-bonding orbital, which on ethylene is empty,
stabilizing the pi structure.

***

This
chapter could cover more, much, much more; and we will pursue more material
along these lines.Especially, our
exploration will heavily revolve around hydrogen and carbon, as well as oxygen
and nitrogen; for these are the main atoms which make up the molecular nature
of life.The basics have been laid down
here.

About Me

My formal training is in chemistry. I also read a great deal of physics and biology. In fact I very much enjoy reading in general, mostly science, but also some fiction and history. I also enjoy computer programming and writing. I like hiking and exploring nature. I also enjoy people; not too much in social settings, but one on one; also, people with interesting or "off-beat" minds draw me to them. I also have some interest in Buddhism.

These days I get a lot more information from the internet, primarily through Wiki. Some television, e. g., documentaries, PBS shows like "Nova" and "Nature".

My favorite science writers are Jacob Bronowski ("The Ascent of Man") and Richard Dawkins (his "The Blind Watchmaker" is right up there up Ascent). I also have a favorite writer on Buddhism, Pema Chodron. Favorite films are "Annie Hall" (by Woody Allen), "The Maltese Falcon", "One Flew Over The Cuckoo's Nest", "As Good As It Gets", "Conspiracy Theory", Monty Python's "Search For The Holy Grail" and "Life of Brian", and a few others which I can't think about at the moment.

I love a number of classical works (Beethoven's "Pastoral", "Afternoon Of A Fawn" and "Clair De Lune" by Debussey , Pachelbel's "Canon" come to mind. My favorite piece is probably Gershwin's "Rhapsody in Blue". But I also enjoy a great deal in modern music, including many jazz pieces, folk songs by people like Dylan, Simon and Garfunkel, a hodgepodge of pieces by Crosby, Stills, and Nash, Niel Young, and practically everything the Beatles wrote.

My life over the last few years has been in some disarray, but I am finally "getting it together.". As I am very much into the sciences and writing, I would like to move more in this direction. I also enjoy teaching. As for my political leanings, most people would probably describe as basically liberal, though not extremely so. My religious leanings are to the absolutely none: I've alluded to my interest in Buddhism, but again this is not any supernatural or scientifically untested aspect of it but in the way it provides a powerful philosophy and set of practical, day to day methods of dealing with myself and the other human beings.